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Recent Advances RECENT ADVANCES THEORY OF VARIATIONAL INEQUALITIES WITH APPLICATIONS TO PROBLEMS OF FLOW THROUGH POROUS MEDIA J. T. ODEN and N. KIKUCHI The University of Texas at Austin, Austin, TX 78712,U.S.A. TABLE OF CONTENTS PREFACE 1173 INTRODUCTItiN’ : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1174 Introductory comments . 1174 Notations and conventions . 1175 1. VARIATIONAL INEQUALITIES . 1177 1.1 Introduction . 1177 1.2 Some preliminary results . 1180 1.3 Projections in Hilbert spaces . 1183 1.4 The Hartman-Stampacchia theorem . 1187 1.5 Variational inequalities of the second kind ’ : : : : : : : . 1188 1.6 A general theorem on variational inequalities . 1189 1.7 Pseudomonotone variational inequalities of the second kind . 1192 1.8 Quasi-variational inequalities . 1194 1.9 Comments . 1201 2. APPROXIMATION AND NUMERICAL ANALYSIS OF VARIATIONAL INEQUALITIES . 1202 2.1 Convergence of approximations . 1202 2.2 Error estimates for finite element approximations of variational inequalities . 1205 2.3 Solution methods . 1210 2.4 Numerical experiments . 1220 3. APPLICATIONS TO SEEPAGE PROBLEMS FOR HOMOGENEOUS DAMS 1225 3.1 Problem setting and Baiocchi’s transformation . 1225 3.2 A variational formulation . 1229 3.3 Special cases . 1237 4. NON-HOMOGENEOUS DAMS . .......... 1246 4.1 Seepage flow problems in nonhomogeneous dams . .......... 1246 4.2 The case k = k(x) . .......... 1247 4.3 Special cases for k = k(x) . .......... 1249 4.4 The case k = k(y) . : : : : : : : : . .......... 1251 4.5 Special cases for k = k(y) . .......... 1255 4.6 Comments . .......... 1256 5. SEEPAGE FLOW PROBLEMS IN WHICH DISCHARGE IS UNKNOWN .......... 1257 5.1 Dam with an impermeable sheet ............... .......... 1257 5.2 Free surface from a symmetric channel .......... 1267 5.3 A seepage flow problem with a horizontal drain 1 1 : : : : 1 : : : .......... 1274 5.4 Comments ....................... .......... 1279 6. CONCLUDING REMARKS ........ .......... 1279 REFERENCES .............. ........ .......... 1280 APPENDIX ............... ........ .......... 1282 References for Appendix .......... ........ .......... 1284 PREFACE Our interest in variational inequalities grew from studies of non-convex optimization theory and pseudomonotone operators as a basis for both the qualitative and numerical analysis of non-linear problems in continuum mechanics. The theory of variational inequalities is rich and exciting; within it, one can find a wealth of powerful ideas which not only reveal fundamental facts on the qualitative behavior of solutions to important classes of non-linear boundary-value problems, but which also provide a natural framework for a host of relatively new numerical methods. Equally important, the theory also enables one to construct a rather elaborate UES Vol. I& No. I&A 1173 1174 J.T. ODEN andN. KIKUCHI approximation theory which brings to light useful information on the behavior of numerical solutions-error estimates, convergence criteria, etc. Finally, at the heart of variational in- equalities is their intrinsic inclusion of free boundaries; thus, they provide a natural and elegant framework for the study of the classical problem of flow through porous media. All of the applications of variational inequalities considered here are focused on problems of this type-the so-called seepage problems of slow irrotational flow of an incompressible fluid through a porous media characterized by Darcy’s law. Our aim in this monograph is to present a rather detailed survey of the theory of variational inequalities, their approximation and numerical analysis, and to demonstrate applications of these theories to the analysis of difficult free boundary problems encountered in the study of flow through porous media. Much of what we discuss here we owe to the principal developers of the subject: Stampacchia, Lions, Biaocchi, Mosco et al., but several of the results we describe, particularly on the computational side, are new. Our account is by no means complete; among other things, we do not treat variational inequalities for evolution problems and we identify several open questions concerning quasi-variational inequalities. We hope that the introduction to these subjects presented here will provide a basis for those who wish to pursue these subjects in more detail. We gratefully acknowledge that the work reported here was completed by the authors during the course of a research project supported by the U.S. National Science Foundation. We also express our thanks to Mrs. Dorothy Baker who skillfully typed the entire manuscript. Austin J. T. ODEN Summer N. KIKUCHI 1979 INTRODUCTION Introductory comments It is a well-known result in convex analysis that the minimization of a functional F defined on a closed convex set K leads to an inequality involving the derivative DF of F rather than the classical equality DF(x) = 0 which is valid when F is defined, for example, on a linear space. This fact has been exploited in the study of convex optimization problems for many years. What was not widely appreciated, however, until a decade ago, was that these ideas had far-reaching implications in many areas of non-linear mechanics; that, in particular, many free-boundary problems could be elegantly formulated using extensions of these ideas, and that, concomitantly, a variety of mathematical methods, both analytical and computational, could be used to study free-boundary problems which were formulated this way. Modern work on the theory of variational inequalities began with the pioneering papers of Fichera[ 11, Stampacchia [2], Lions and Stampacchia[3] and Brezis [4], and was further developed by the French and Italian school of applied mathematicians during the last decade (e.g. Mosco[5], Glowinski, Lions and Tremolieres [6], Fichera[7], Duvaut and Lions[8]). Excellent surveys of these ideas have been contributed by Mosco[5,9], Stampachia[lO] and Lions [ 1I]; applications to a wide variety of free-boundary problems are discussed in the book of Duvaut and Lions[8]. Numerical methods based on variational inequalities are discussed in the two-volume text of Glowinski, Lions and TrCmolieres[6] and in the monograph of Glowinski[ 121.The application of variational inequalities to free-boundary problems arising in the flow of fluids through porous media was studied by Baiocchi[l3] and Baiocchi et al.[14], and a numerical analysis of such problems was investigated by Baiocchi et al. [ IS]. Theorems on the convergence of finite element approximation of certain classes of variational inequalities were developed by Falk[l6], Brezzi, Hager and Raviart [17] et al. Additional references to literature on variational inequalities and their applications can be found in the works cited above. We will also cite other references relevant to our study later in this work. Our objective here is five-fold: 1. To give a summary account of the general mathematical theory of variational inequalities set in the framework of non-linear operators defined on convex sets in real Banach spaces. We focus our attention on existence and uniqueness theorems for such abstract problems for, as Theory of variational inequalities, flow through porous media 1175 will be shown, these form the basis for the construction and analysis of numerical methods for such problems. 2. To study the approximation of variational inequalities by finite element methods, and to study various numerical schemes that can be used to solve discrete models of variational inequalities. 3. To describe the formulation of the seepage flow problem by variational inequalities using variants of the Baiocchi transformation, and to study the existence and regularity of solutions to such problems. 4. To develop finite element methods for the approximate solution of seepage flow prob- lems. Here we are also concerned with the existence of solutions to the approximate problems, the convergence of finite element approximations, and the development of a priori error estimates. 5. To solve numerically several representative seepage flow problems and to discuss and compare various numerical schemes. The theoretical foundations of variational inequalities are taken up in Chap. 1 following this introduction. There we give a rather complete account of the theory as it applies to monotone and pseudomonotone operators on reflexive Banach spaces. We also discuss the theory of quasi-variational inequalities, which we later show to be very important in the study of certain seepage problems. Finite element approximations and various numerical methods are discussed in Chap. 2. We review the theory of Falk[l6] for error estimation of certain classes of variational inequalities, and we describe algorithms for the solution of systems of inequalities; in particular, we examine fixed-point (contraction mapping) methods, S.O.R.-projection methods, Lagrange multiplier methods, and penalty methods. Some numerical experiments designed to test the validity of the theoretical estimates and to compare methods are also presented in this section. For completeness, we give proofs to all of the major theorems discussed in Chaps. 1 and 2. The formulation of seepage flow problems is discussed in Chap. 3. Here a rather general formulation is developed, using the notion of quasi-variational inequalities. We then consider a number of special cases, describe some numerical experiments, and compare results with those obtained by other methods. Chapter 4 is devoted to the analysis of seepage in non-homogeneous dams in which the permeability k at a point (x, y)
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